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For example, on $\mathbb{R}$ there exists trivial topology which contains only $\mathbb{R}$ and $\emptyset$ and in that topology all open sets are closed and all closed sets are open.

Question. Does there exist non-trivial topology on $\mathbb{R}$ for which all open sets are closed and all closed sets are open? Further, given some general set $X$ whose number of elements is finite, could we always construct non-trivial topology where all open sets are closed and all closed sets are open? What if $X$ has a non-finite number of elements?

I hope my question is not meaningless.

Thank you for any help.

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    $\begingroup$ The trivial topology on a set $X$ is $\{\emptyset, X\}$. The discrete topology on a set $X$ is $2^X$. Both have the property that you are looking for, though I am not sure if you are using the term "trivial" here in reference to the trivial topology or just to mean "simple". $\endgroup$ – parsiad Mar 17 '19 at 1:04
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    $\begingroup$ (1) The discrete topology, where all subsets are open and closed. (2) Given any partition $\{X_i\}_{i\in I}of $X$, take as a topology the colleciton of all sets that are unions of elements of the partition; the complements are also unions of elements of the partition. This works for any set, regardless of cardinality, and any partition, regarless of size of the partition or of its constituent sets. $\endgroup$ – Arturo Magidin Mar 17 '19 at 1:05
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    $\begingroup$ Related to your question are door spaces and extremally disconnected spaces. $\endgroup$ – William Elliot Mar 17 '19 at 1:08
  • $\begingroup$ As noted in the answers, the spaces you are asking about are effectively discrete. The extremally disconnected spaces (spaces such that the closure of any open set is open) suggested by @WilliamElliot can be much more interesting. $\endgroup$ – tomasz Mar 17 '19 at 13:48
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The topologies with this property are precisely the ones that are derived from partitions of the underlying set $X$.

Specifically, let $\mathcal{P}=\{X_i\}_{i\in I}$ be a partition of $X$, and let $\tau$ be the collection of sets of the form $\cup_{i\in I_0}X_i$ for $I_0\subseteq I$. Then $\tau$ is a topology: the empty set corresponds to $I_0=\varnothing$, the set $X$ to $I_0=I$; the union of such sets corresponds to the family indexed by the union of indices, and the intersection to the intersection of the indices. Moreover, the complement of the set corresponding to $I_0$ is the set corresponding to $I-I_0$. Thus, $\tau$ has the desired property.

Now let $\tau$ be any topology on $X$ with the desired property. Define an equivalence relation on $X$ by letting $x\sim y$ if and only if for every $A\in \tau$, $x\in A$ if and only if $y\in A$. Trivially, this is an equivalence relation, and so induces a partition on $X$. I claim that $\tau$ is in fact, the topology induced by this partition as above.

Indeed, if $A\in \tau$, then $A=\cup_{x\in A}[x]$, where $[x]$ is the equivalence class of $x$. Trivially $A$ is contained in the right hand side, and if $y\in[x]$, then since $x\in A$ then $y\in A$, so we have equality.

Now, conversely, let $x\in X$ and look at $[x]$. I claim that $X-[x]$ lies in $\tau$. So see this, let $z\in X-[x]$. Then since $z\notin [x]$, there exists an open set $A_z\in \tau$ such that $z\in A_z$ but $x\notin A_z$ (and hence, $[x]\cap A_z=\varnothing$). Now, $\cup_{z\notin[x]}A_z$ is an open set, contains every element of $X-[x]$, and intersects $[x]$ trivially because each element in the union does. That is, this open set is $A-[x]$; but since the complement of every open set is open, and $A-[x]$ is open, then $[x]$ is open. Thus, $[x]\in\tau$.

We have then proven that every element of the partition induced by $\sim$ is open, and that every open set is a union of such elements of the partition. That is, the open sets are precisely the unions of elements of the partition $X/\sim$.


Added. The equivalence relation given in the second part can be defined in any topology, of course; and the proof that every open set is a union of equivalence classes and that the complement of an equivalence class is open, always hold. The only place where we used the hypothesis that all open sets are closed was to conclude the equivalence class itself was open. For example, in the standard topology for $\mathbb{R}$, the equivalence relation is the trivial one.

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You can manufacture examples like this pretty easily. Let $A$ be any subset of $\mathbb{R}$. Put a topology on $\mathbb{R}$ with the following open sets: $$ \varnothing, A, \mathbb{R}-A, \mathbb{R}. $$ You can easily check that this always gives a topology, and a subset of $\mathbb{R}$ is open if and only if it is closed. This construction generalizes to any set $X$: there's nothing special about $\mathbb{R}$ here.

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  • $\begingroup$ More generally, take any partition and take unions of elements of the partition. This example is what you get from the partition $\{A,\mathbb{R}-A\}$. $\endgroup$ – Arturo Magidin Mar 17 '19 at 1:05
  • $\begingroup$ @ArturoMagidin Let us denote with X set of all topologies which satisfy property in the question on $\mathbb{R}$. Can every topology in $X$ be made with some partitions? $\endgroup$ – Thom Mar 17 '19 at 1:14
  • $\begingroup$ @Thom: Yes. I’ll write it up. $\endgroup$ – Arturo Magidin Mar 17 '19 at 1:51
  • $\begingroup$ @ArturoMagidin Fascinating. Thank you. $\endgroup$ – Thom Mar 17 '19 at 1:59

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